Díaz Báñez, José MiguelGarijo Royo, DeliaMárquez Pérez, AlbertoUrrutia Galicia, Jorge2017-05-232017-05-232013Cibulka, J., Korbelář, M., Kynčl, J., Mészáros, V., Stolař, R. y Valtr, P. (2013). On three parameters of invisibility graphs. En XV Spanish Meeting on Computational Geometry, Sevilla.http://hdl.handle.net/11441/60268The invisibility graph I(X) of a set X ⊆ Rd is a (possibly infinite) graph whose vertices are the points of X and two vertices are connected by an edge if and only if the straight-line segment connecting the two corresponding points is not fully contained in X . We consider the following three parameters of a set X : the clique number ω(I(X)), the chromatic number χ(I(X)) and the inimum number γ(X) of convex subsets of X that cover X. We settle a conjecture of Matousek and Valtr claiming that for every planar set X, γ(X) can be bounded in terms of χ(I(X)). As a part of the proof we show that a disc with n one-point holes near its boundary has χ(I(X)) ≥ log log(n) but ω(I(X)) = 3. We also find sets X in R5 with χ(I(X)) = 2, but γ(X) arbitrarily large.application/pdfengAttribution-NonCommercial-NoDerivatives 4.0 Internacionalhttp://creativecommons.org/licenses/by-nc-nd/4.0/info:eu-repo/semantics/conferenceObjectinfo:eu-repo/semantics/openAccess